Shôichirô Sakai was born in 1928 in Kanuma, Japan. He earned his B.A. in mathematics in 1953 and his doctorate in 1961 from Tohoku University. He worked as a research assistant there from 1953 to 1960, then became a faculty member at Waseda University (1960–1964). After two years at the University of Pennsylvania, he became a professor there in 1966 and remained until 1979, when he returned to Japan to take a chair at Nihon University. He also held visiting positions at Yale University (1962–64) and MIT (1967–68). His main interests are in operator algebras, functional analysis, and mathematical physics. This book is a reprint of the 1971 edition of Sakai's work on C*-algebras and W*-algebras. It was originally published as Vol. 60 of the Ergebnisse der Mathematik und ihrer Grenzgebiete. The book discusses the theory of operator algebras in Hilbert spaces, initiated by von Neumann in 1929. The theory has become a common tool in various fields of mathematics and theoretical physics. The book covers two main approaches to studying operator *-algebras: W*-algebras (weakly closed) and C*-algebras (uniformly closed). It includes a detailed discussion of the classification and representation theory of W*-algebras, reduction theory, and special topics such as derivations, automorphisms, and type I C*-algebras. The book also includes a Stone-Weierstrass theorem for C*-algebras. The author thanks various individuals and institutions for their support during the preparation of the book. The book is divided into chapters covering general theory, classification and representation theory of W*-algebras, reduction theory, and special topics. It includes a bibliography and subject index.Shôichirô Sakai was born in 1928 in Kanuma, Japan. He earned his B.A. in mathematics in 1953 and his doctorate in 1961 from Tohoku University. He worked as a research assistant there from 1953 to 1960, then became a faculty member at Waseda University (1960–1964). After two years at the University of Pennsylvania, he became a professor there in 1966 and remained until 1979, when he returned to Japan to take a chair at Nihon University. He also held visiting positions at Yale University (1962–64) and MIT (1967–68). His main interests are in operator algebras, functional analysis, and mathematical physics. This book is a reprint of the 1971 edition of Sakai's work on C*-algebras and W*-algebras. It was originally published as Vol. 60 of the Ergebnisse der Mathematik und ihrer Grenzgebiete. The book discusses the theory of operator algebras in Hilbert spaces, initiated by von Neumann in 1929. The theory has become a common tool in various fields of mathematics and theoretical physics. The book covers two main approaches to studying operator *-algebras: W*-algebras (weakly closed) and C*-algebras (uniformly closed). It includes a detailed discussion of the classification and representation theory of W*-algebras, reduction theory, and special topics such as derivations, automorphisms, and type I C*-algebras. The book also includes a Stone-Weierstrass theorem for C*-algebras. The author thanks various individuals and institutions for their support during the preparation of the book. The book is divided into chapters covering general theory, classification and representation theory of W*-algebras, reduction theory, and special topics. It includes a bibliography and subject index.